Following the recent proof I co-authored in Mathematics, I finally give here a pure Peano-arithmetical demonstration that all Collatz orbits converge to 1. The point is to establish that the respective attractors of any pair of odd numbers, if we assumed they were separate, could always be finitely proven to collide upward, which is done by demonstrating they branch faster than the binary tree over odd numbers. This had me introduce two novel frameworks: Romanesco algebra and Dreamcatcher theory, which I discuss, along with Tao's fine-tuned "epsilon management" or what could then be called Poincaré-Denjoy-Tao theory, could crack other Collatz-like problems (e.g. the Juggler sequence). Here I just discuss how it could be used to independently prove there are no non-trivial cycles in Collatz, using only what Tao already had, and provide a new perspective on the Furstenberg ×2 ×3 conjecture.